Scientific Computing Partial Differential Equations Introduction and Finite Difference Formulas

Partial Differential Equations A partial differential equation (PDE) is an equation involving partial derivatives of an unknown function of two or more independent variables The following are examples. Note: u may depend on spatial variables and possibly a time variable.

Partial Differential Equations

Partial Differential Equations Because of their widespread application in engineering, our study of PDE will focus on linear, second-order equations The following general form will be evaluated for B2 - 4AC (Variables – x and y/t)

Partial Differential Equations B2-4AC Category Example < 0 Elliptic Laplace equation (steady state with 2 spatial dimensions) = 0 Parabolic Heat conduction equation (time variable with one spatial dimension) >0 Hyperbolic Wave equation (time-variable with one spatial dimension)

Heat Equation One of the classical partial differential equation of mathematical physics is the equation describing the conduction of heat in a solid body (Originated in the 18th century). Consider a straight bar with uniform cross-section and homogeneous material. We wish to develop a model for heat flow through the bar.

Heat Equation Let u(x,t) be the temperature on a cross section located at x and at time t. We shall follow some basic principles: A. Fourier’s Law: The amount of heat energy per unit time flowing through a unit of cross-sectional area is proportional to with constant of proportionality k(x) called the thermal conductivity of the material.

Heat Equation B. Heat flow is always from points of higher temperature to points of lower temperature. C. The amount of heat energy necessary to raise the temperature of an object of mass “m” by an amount u is a “c(x) m(x) u”, where c(x) is the specific heat capacity of the material.

Heat Equation By Fourier’s Law the amount of heat H(x) flowing from left to right through the surface A of a cross section at x during the time interval t can be approximated by: A x

Heat Equation Likewise, the amount of heat H(x+ x) flowing from left to right through a cross section at (x + x) during the time interval t is about: B x + x

Heat Equation Then, on the interval [x, x+x], during time t , the total change in heat is approximately: A B x x + x

DH = c.m.Du=c(x) .((x) Dx).Du Heat Equation Dividing by t x we get: From Item C above for a change in x of Dx : DH = c.m.Du=c(x) .((x) Dx).Du where (x) is the linear mass density function.

Heat Equation Thus: Substituting this into the formula from the previous slide gives:

Heat Equation Canceling x on the left we get: If we take the limits as t, x  0, we get:

Heat Equation If we assume k, c,  are constants, then the eq. becomes: ( ) where This is the Heat Equation in one (space) dimension.

Boundary and Initial conditions We need to designate what the initial temperature distribution is in the rod: u(x,0) We also need to designate what the temperature function is at the ends of the rod: u(0,t) and u(L,t) where L = length of rod. For example, if the ends of the rod are kept at constant temps T1 and T2 ,then u(0,t) = T1 and u(L,t) = T2

One Dimensional Heat Equation

Multi-dimensional space Now consider an object in which the temperature is a function of more than just the x-direction. Then the heat conduction equation can then be written: 2-D: 3-D:

Solving the Heat Equation A solution u(x,t) for the heat equation is a function that satisfies the PDE and all initial conditions. Solution methods: Method of Finite Differences (MFD) Method of Finite Elements (MFE)

Finite Differences The method of finite differences approximates the value of the derivatives of u(x,t) at a point (x0,t0) in its domain, say by using a combination of function values at nearby points. Method is due to Newton. Start with simpler case of f(x)

Differentiation The mathematical definition of a derivative begins with a difference approximation: and as x is allowed to approach zero, the difference becomes a derivative:

Differentiation Formulas Taylor series expansion can be used to generate high-accuracy formulas for derivatives by using the expansion around several points around a given point xi. Three categories for the formula include forward finite-difference backward finite-difference centered finite-difference

Differentiation Formulas Forward difference Backward difference Centered difference

Forward Difference True derivative Approximation h x xi1 xi xi+1

Backward Difference True derivative Approximation h x xi1 xi xi+1

Centered Difference True derivative Approximation 2h x xi1 xi xi+1

First Derivatives x i-2 i-1 i i+1 i+2 Forward difference Backward difference Central difference

Second Derivatives Using the Taylor series expansion about xi we get: where Thus, And,

Centered Finite-Difference

Forward Finite-Difference

Backward Finite-Difference